Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [calculus]

For basic questions about limits, continuity, derivatives, differentiation, integrals, and their applications, mainly of one-variable functions.

Calculus is the branch of mathematics studying the rate of change of quantities, which can be interpreted as slopes of curves, and the lengths, areas and volumes of objects.

Calculus is divided into differential and integral calculus, which are concerned with derivatives

$$\frac{\mathrm{d}y}{\mathrm{d}x}= \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x}$$

and integrals

$$\int_a^b f(x)\,\mathrm{d}x = \lim_{\Delta x \to 0} \sum_{k=0}^n f(x_k)\ \Delta x_k,$$

respectively.

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Motivation For Biology Students

Can someone give me ideas for specific examples that might motivate biology/chemistry students to learn basic calculus ( limits, derivatives and basic integrals and theorems such as Lagrange's, Rolle's ,etc... ) Thanks in advance !
joshua
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Explicit components of $y^x=x^y$

When graphing the implicit equation $y^x=x^y$ it seems like it consists of two possibly explicit curves. On the one hand there is the linear solution $y=x$. But as it appears there is a second curve monotonously decreasing as x increases,…
A.Pz
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Bolzano–Weierstrass theorem conclusion

Can I conclude from Bolzano–Weierstrass theorem that there is more than one convergent subsequence, or the theorem tells me that there's only one ? To be more clear, given a bounded sequence $X_n$, not ecessarily converges, can I conclude there are…
Itay4
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How do you differentiate the following function?

How do you take the derivative of the following function: $$y=\ln(|\sec(5x) + \tan(5x)|)$$ So $dy/dx$ of the following function. Thanks in advance! The steps I have taken don't seem to be correct: $$y=\ln(|\sec(5x) + \tan(5x)|)$$ $$y'={\frac{\frac…
Sam Liemburg
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Let $f$ be continuous on $ [a,b]$, differentiable on $ (a,b) $ and $ f(x) \neq 0 \forall x \in (a,b) $ Prove $ \exists c \in (a,b) $ such that

To Prove : $$ \frac{f'(c)}{f(c)} = \frac{1}{a-c} + \frac{1}{b-c} $$ I think I should proceed in following way : Define $g(x) = \ln|f(x)| + \ln|x-a| + \ln|b-x|$ such that $$g'(x) = \frac{f'(x)}{f(x)} - \frac{1}{a-x} - \frac{1}{b-x} $$ Then, using…
Sarvesh Nalawade
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real values of p in equation involving floor function

if $\displaystyle \sin \alpha = p \bigg\lfloor \int^{1}_{0}\{\ln x\}dx\bigg\rfloor \;, \alpha \in (0,2\pi)$ .Then $p$ is ? given $\lfloor x \rfloor $ is floor function of $x$ and $\{x\} = x-\lfloor x \rfloor$ using $\{\ln x \} = \ln (x) - \lfloor…
DXT
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Use Mean value theorem to prove the following inequality

A) Use the Mean value theorem to prove that \begin{equation} \sqrt{1+x} < 1 + \frac{1}{2}x \text{ if } x>0 \end{equation} B) Use result in A) to prove that \begin{equation} \sqrt{1+x}>1+\frac{1}{2}x-\frac{1}{8}x^2 \text{ if } x>0 \end{equation} Can…
Little Rookie
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Can a function be differentiable at only isolated points?

It is possible for a derivative to fail to exist at isolated points, but I would like to know if a function could be constructed that is not differentiable almost everywhere and differentiable at isolated points only.
Michael Stachowsky
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Adding constant to the variable in a Limsup

Let $f(t)$ be a real valued function and $C$ a constant. Is it true that $$ \limsup_{t\to\infty}\frac{f(t)}{t}=\limsup_{t\to\infty}\frac{f(t+C)}{t} ? $$ I have tried to prove it using a change of variable $s=t+C$, but the denominator seems…
Sak
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Question on calculus

I want to compute $\displaystyle{\sum_{n=4}^{\infty}{1 \over n^{3} + n^{2}\cos\left(n\right)}.\quad}$ Can anyone help me or give a hint?
Eric Zhao
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Derivative of an expression

I completely forgot how to find derivatives, can someone give me an example of a simple equation and how to find its derivative? The only thing I remember is that the formula is$$f'(x)=\frac{f(x+h)-f(x)}{h}$$ Thanks for the help.
Randomizer
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Cantor Set Geometric Mean

Find the Geometric Mean of all reals existing as part of the Cantor Set between (0,1]. I've been trying to solve this problem, but keep messing up the sets I construct for higher iterations. Any help would be…
user245640
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Clarification on what this author means by "logging does not change the maximum" of a function

I am self-studying for an actuarial exam and I encountered the following: The author seems to suggest that if we want to find the maximum of a function $f(x)$ with respect to $x$: We can drop any multiplicative constant. We can take the natural…
Joseph DiNatale
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A question about differentiability

Does there exist a continuous function $f:\mathbb{R}\to\mathbb{R}$ so that $f$ is differentiable exactly at one point?
Chung
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Proof: Antiderivative of a function differ by a constant

Let $f(x)$ a function defined at $I\subseteq \mathbb{R}$ and assume that $F(x)$ and $G(x)$ are the antiderivatives of $f(x)$ in $I$, so there is a $c$ such that for all $x\in I$, $F(x)=G(x)+c$ Let us define $H(X)=F(x)-G(X)$ therefore…
gbox
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